On First-Order Expressibility of Satisfiability in Submodels

Let kappa, lambda be regular cardinals, lambda <= kappa, let phi be a sentence of the language L-kappa,L-lambda in a given signature, and let v(phi) express the fact that phi holds in a submodel, i.e., any model in the signature satisfies v(phi) if and only if some submodel B of U satisfies phi. It was shown in [1] that, whenever phi is in L-kappa,L-omega in the signature having less than kappa functional symbols (and arbitrarily many predicate symbols), then v(phi) is equivalent to a monadic existential sentence in the second-order language L-kappa,omega(2), and that for any signature having at least one binary predicate symbol there exists phi in L-omega,L-omega such that v(phi) is not equivalent to any (first-order) sentence in L-infinity,L-omega. Nevertheless, in certain cases v(phi) are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when v(phi) is in L-kappa,L-kappa and kappa is omega or a certain large cardinal.